#315762

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

671 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,12,16,14)(17,18,22,24)(19,27,28,25)(20,23,29,32)(21,30,31,26);(1,3)(2,5)(4,7)(6,8)(10,15)(11,16)(17,19,21,20,22,28,31,29)(18,23,26,27,24,32,30,25);(3,7)(5,8)(9,11,14,10,13,16,12,15)(18,24)(23,32)(25,27)(26,30);(9,12,13,14)(10,11,15,16)(17,20,21,28,22,29,31,19)(18,25,26,23,24,27,30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=ei,c^2=fh,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222958);